Question 1

A rectangular fence is to be constructed using 200 ft of fencing, with one side formed by a rock wall that does not require fencing. What are the dimensions that maximize the area enclosed?

Accepted Answer

Width $$= 50 ft$$, Length $$= 100 ft$$

Question 2

A shipping company wants to construct a rectangular box with a square base and no lid, using the least amount of cardboard for a volume of $$125 in$^3. $$What is the formula for the surface area in terms of the base side $$x$$?

Accepted Answer

$$A = 4x \cdot y + x^2$

Question 3

For the box in the previous question, if $$y = \frac{125}{x^2$}$$, what is the surface area function $$A(x)$$?

Accepted Answer

$$A(x) = 4x \cdot \frac{125}{x^2$} + x^2$

Question 4

A poster has a total area of $$1150 cm$^2$$, with $$2-cm $$margins on the sides and top, and a $$3-cm $$margin at the bottom. What is the formula for the printed area in terms of $$x$$?

Accepted Answer

$$A_p = (x-4)(\frac{1150}{x} - 5)$$

Question 5

A coffee shop's price-demand function is $$p(x) = 100 - \frac{x}{4}. $$What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = (100 - \frac{x}{4})x$$

Question 6

Given $$R(x) = 100x - \frac{x^2$}{4}$$, what is the domain restriction for maximizing revenue?

Accepted Answer

$$0 \u003c x \u003c 400$$

Question 7

Gerald and Katya sell custom phone cases. The price per case is $$p(x) = 50 - 0.2x$$ and the cost is $$C(x) = 100 + 15x. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = (50 - 0.2x)x - (100 + 15x)$$

Question 8

Your café sells lattes for $$4$$ each to $$100$$ customers per day. For every $$1$$ increase in price, you lose $$20$$ customers. What is the revenue function $$R(p) in $$terms of price $$p$$?

Accepted Answer

$$R(p) = p \cdot (100 - 20(p-4))$$

Question 9

A Norman window consists of a rectangle with width $$x$$ and height $$y$$, topped by a semicircle of radius $$\frac{x}{2}. If $$the perimeter is $$24$$ meters, what is the perimeter equation?

Accepted Answer

$$P = 2y + x + \frac{\pi x}{2}$$

Question 10

For the Norman window, if the rectangular part lets in four times as much light per square meter as the semicircular part, what is the function to maximize?

Accepted Answer

$$L = xy + \frac{1}{4} \cdot \frac{\pi x^2$}{4}$$

Question 11

A box with an open top is formed by cutting equal-sized squares from the corners of a $$3 ft by$$ $$8 ft $$sheet. What is the volume function $$V(x) in $$terms of the side $$x of $$the cut squares?

Accepted Answer

$$V(x) = x(8-2x)(3-2x)$$

Question 12

For the box in the previous question, what is the domain restriction for $$x$$?

Accepted Answer

$$0 \u003c x \u003c 1.5$$

Question 13

A $$10-$$meter-tall tree stands $$5$$ meters from a cliff. What is the length of the shortest rope that can be tied from the ground over the top of the tree to the top of the cliff?

Accepted Answer

The minimum occurs when the rope is tangent to the tree, requiring optimization using calculus.

Question 14

When solving an applied optimization problem, what is the first step?

Accepted Answer

Draw a diagram and identify variables

Question 15

In applied optimization, how do you determine if a critical point is a maximum or minimum when there are no endpoints?

Accepted Answer

Plug the critical point into the second derivative; negative means maximum, positive means minimum

Question 16

Which of the following is NOT a typical step in solving an applied optimization problem?

Accepted Answer

Write the function in terms of two variables

Applied Optimization in Business Calculus

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